The present application relates to investment portfolios and, more particularly, to management of risk and allocation of assets in an investment portfolio. Merely by way of example, the invention is illustrated in a computing environment. But it would be recognized that the invention has a much broader range of applicability.
The evaluation and management of risk in a portfolio of assets continue to be a subject of interest by many professionals in the field of finance. The investment objectives, in terms of rates of return as well as levels of risk associated with such returns, often vary for different individuals. Furthermore, the array of factors that affect investment objectives do not easily lend themselves to simple computational algorithms. Accordingly, there appear to be many limitations with conventional techniques.
To reduce or possibly minimize risk while providing an acceptable rate of return, an investment portfolio is typically diversified so as to include various assets with different characteristics. Portfolio construction was generally reshaped in the 1950s, when the concept of portfolio theory was developed (see “Portfolio Selection”, Journal of Finance 7, no. 1, March 1952, 77–91, for example). The portfolio theory—which in the field of finance is also referred to as mean variance analysis—seeks to balance the reward from the portfolio against the risk that the portfolio is exposed to, where risk is measured by the variance or standard deviation of portfolio return. The portfolio theory assumes that over the investment horizon, an array of assets—of different classes—follow a multi-variate probability distribution with finite expected value vector and covariance matrix. A portfolio constructed in conformity with the theory combines such assets in a linear fashion so as to achieve a single-dimensional probability distribution of investment returns which provides for either (1) maximum expected return for a given standard deviation (i.e., a given level of risk) or (2) a minimum standard deviation for a given expected return, or (3) maximum expected return for a given tradeoff between expected return and risk. By computing a probability distribution function of portfolio return, the theory thus provides as mathematical foundation for constructing optimal portfolios.
A plot of an expected portfolio return as a function of the standard deviation of portfolio return, also known as the efficient frontier, is shown in the simplified diagram of FIG. 1. The efficient frontier provides the maximum return for every given level of risk, or alternatively, provides the minimum risk for every level of return. Therefore, optimal portfolios often lie on the efficient frontier. To construct an efficient frontier, in accordance with the portfolio theory, the expected return and the standard deviation of the return for each asset in the portfolio, as well as, the correlation between the returns of each asset in the portfolio are estimated for a given time period. A computer software that implements the technique, typically receives the required data and generates an optimum investment weigh for each asset in the portfolio.
In mean-semivariance analysis, which is another known portfolio construction theory, the semi-variance—which is a measure of downside risk—of portfolio return is used in portfolio construction as the measure of risk. In accordance with this theory, only portfolio returns below a specified level are used in calculation of the efficient frontier plots. Like its mean-variance counterpart, the mean-semivariance uses a one-period approach for construction and optimization of a portfolio of diverse assets.
Another known method uses a power utility function to construct a portfolio. The method (hereinbelow referred to as the power-utility method) maximizes the expected utility of wealth—which is related to portfolio return—using a multi-period model which spans the investment horizon. Because the power-utility method is based on a multi-period model, it allows for modification and rebalancing of the portfolio during each period, in contrast to the mean-variance and mean-semivariance methods which are based on a one period model. In addition to the mean and the variance—which are the first two moments of a probability distribution function—that are used in the mean-variance model, described briefly above, the power-utility method uses the higher orders moments of a probability distribution function to determine the effect of a portfolio on an investor's wealth over the investment horizon. Such higher order moments include preferences for e.g. skewness and kurtosis or fat-tails.
The power utility method (see “Higher Return, Lower Risk: Historical Returns on Long-Run, Actively Managed Portfolios of Stock, Bond and Bills, 1936–1978” by Robert R. Grauer and Nils H. Hakansson, Financial Analysts Journal, pp. 39–53, March–April 1982) is based on the following power utility function:
  U  =            1      γ        ⁡          [                                    (                          1              +              r                        )                    γ                +        γ        -        1            ]      in which                U represents portfolio's utility to the portfolio holder;        r represents the portfolio's return; and        γ represents the risk-aversion of the portfolio holder and has a value of less than or equal to 1.        
A property—commonly referred to as the myopic property—of the power utility function is that it lends itself to maximization of the expected utility of wealth at the end of each period of a given multi-period investment horizon, thereby, providing for the maximization of the expected utility at the end of investment horizon.
For an individual who is risk-neutral, γ is 1. Values of γ less than one, including negative values, represent risk-averse investors. The higher the risk aversion, the more negative is the parameter γ. Therefore, γ values of 0.5, 0, −1, −5, −15, respectively represent greater aversion to both decreases and increases in the portfolio value and hence to the risk associated with the portfolio.
FIG. 1 shows a simplified diagram of an investor's utility (y-axis) as a function of (1+portfolio return) (x-axis), where (1+portfolio return) is a measure of the growth in portfolio value, or wealth. Plot A in FIG. 1 corresponds to a power-utility function for an individual with a γ of −15. As seen from this plot, a portfolio constructed in accordance with graph A, provides substantial protection against negative returns—due to its steep drop in utility as the rate of return drops—at the expense of limiting upside potential by providing low positive returns. In other words, a portfolio corresponding to plot A, immunizes the downside exposure of the portfolio while at the same time suppressing the upside potential of the portfolio.
The power utility function simplifies to the following log-utility function when γ is equal to 0:Us=1+1n(1+r)in which 1n represents the symbol for natural logarithm.
Plot B of FIG. 1 shows the utility derived from a portfolio constructed using a log-utility function. As is well know, a portfolio selected by using the log-utility function maximizes growth over multiple periods, but as described below, such a portfolio may be very risky. Plots A and B of FIG. 1 show that the portfolio whose utility is characterized by plot B provides a higher utility at higher positive rates of returns than does the portfolio whose utility is characterized by plot A. However, the negative utility of the portfolio whose utility is characterized by plot B is not as suppressed as is the portfolio whose utility is characterized by plot A. Therefore, a portfolio whose utility is characterized by plot B is subject to a higher risk of loss than is a portfolio whose utility is characterized by plot A.
The enhanced risk of loss of the log-utility function is the by-product of the well-known fact that portfolios constructed based thereon often contain riskier assets and higher investment weight of such riskier assets, than those constructed using the power-utility functions with large negative values of γ. Therefore, although a portfolio based on a log-utility function offers a higher potential for positive returns than does a portfolio based on a power-utility function with a large negative γ, a portfolio based on a log-utility function is subject to a higher potential negative return than is a portfolio based on a power-utility function with a large negative γ.
Therefore, it is desirable to construct a diversified portfolio model that is more efficient than conventional techniques.